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Multiscale radial kernels with high-order generalized Strang-Fix conditions

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Abstract

The paper provides a general and simple approach for explicitly constructing multiscale radial kernels with high-order generalized Strang-Fix conditions from a given univariate generator. The resulting kernels are constructed by taking a linear functional to the scaled f -form of the generator with respect to the scale variable. Equivalent divided difference forms of the kernels are also derived; based on which, a pyramid-like algorithm for fast and stable computation of multiscale radial kernels is proposed. In addition, characterizations of the kernels in both the spatial and frequency domains are given, which show that the generalized Strang-Fix condition, the moment condition, and the condition of polynomial reproduction in the convolution sense are equivalent to each other. Hence, as a byproduct, the paper provides a unified view of these three classical concepts. These kernels can be used to construct quasi-interpolation with high approximation accuracy and construct convolution operators with high approximation orders, to name a few. As an example, we construct a quasi-interpolation scheme for irregularly spaced data and derived its error estimates and choices of scale parameters of multiscale radial kernels. Numerical results of approximating a bivariate Franke function using our quasi-interpolation are presented at the end of the paper. Both theoretical and numerical results show that quasi-interpolation with multiscale radial kernels satisfying high-order generalized Strang-Fix conditions usually provides high approximation orders.

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References

  1. Bozzini, M., Dyn, N., Rossini, M.: Construction of generators of quasi-interpolation operators of high approximation orders in spaces of polyharmonic splines. J. Comput. Appl. Math. 236, 577–564 (2011)

    MathSciNet  MATH  Google Scholar 

  2. Bozzini, M., Rossini, M., Schaback, R.: Generalized Whittle-Matérn and polyharmonic kernels. Adv. Comput. Math. 39, 129–142 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Bozzini, M., Rossini, M., Schaback, R., Volontè, E.: Radial kernels via scale derivatives. Adv. Comput. Math. 41, 277–291 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Bozzini, M., Lenarduzzi, L., Rossini, M., Schaback, R.: Interpolation with variably scaled kernels. IMA J. Numer. Anal. 35, 199–219 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Buhmann, M.: Radial functions on compact support. Pocced. Edinburgh. Math. Soci. 41, 33–46 (1988)

    MathSciNet  MATH  Google Scholar 

  6. Buhmann, M.: Multivariable Interpolation Using Radial Basis Functions, Ph.D. dissertation, University of Cambridge (1989)

  7. Buhmann, M.: Radial basis functions. Acta Numer. 1, 1–38 (2000)

    MathSciNet  MATH  Google Scholar 

  8. Buhmann, M.: Radial Basis Functions: Theory and Implementations, Cambridge Monographs on Applied and Computational Mathematics. Cambridge Univ. Press, Cambridge (2003)

    Google Scholar 

  9. Cheney, E., Light, W., Xu, Y.: On kernels and approximation orders, Approximation Theory (Memphis, TN). Lecture Notes in Pure and Applied Mathematics, Dekker, NewYork 138, 227–242 (1992)

    MATH  Google Scholar 

  10. Dyn, N., Jackson, I., Levin, D., Ron, A.: On multivariate approximation by integer translates of a basis function. Israel J. Math. 78, 95–130 (1992)

    MathSciNet  MATH  Google Scholar 

  11. Fasshauer, G.E.: Meshfree Approximation Methods with MATLAB. World Scientific Publishers, Singapore (2007)

    MATH  Google Scholar 

  12. Fornberg, B., Flyer, N.: Solving PDEs with radial basis functions. Acta Numer. 24, 215–258 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Fornberg, B., Larsson, E., Wright, G.: A new class of oscillatory radial basis functions. Comput. Math. Appl. 51, 1209–1222 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Gao, W.W., Wu, Z.M.: Quasi-interpolation for linear functional data. J. Comput. Appl. Math. 236, 3256–3264 (2012)

    MathSciNet  MATH  Google Scholar 

  15. Gao, W.W., Wu, Z.M.: A quasi-interpolation scheme for periodic data based on multiquadric trigonometric B-splines. J. Comput. Appl. Math. 271, 20–30 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Gao, W.W., Wu, Z.M.: Constructing radial kernels with higher-order generalized Strang-Fix conditions. Adv. Comput. Math. 43, 1355–1375 (2017)

    MathSciNet  MATH  Google Scholar 

  17. Gao, W.W., Zhang, R.: Multiquadric trigonometric spline quasi-interpolation for numerical differentiation of noisy data: a stochastic perspective. Numer. Algor. 77, 243–259 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Gao, W.W., Sun, X.P., Wu, Z.M., Zhou, X.: Multivariate Monte Carlo approximation based on scattered data, submitted

  19. Gao, W.W., Fasshauer, G.E., Sun, X.P., Zhou, X.: Optimality and regularization properties of quasi-interpolation: deterministic and stochastic approaches, submitted

  20. Jia, R.Q., Lei, J.J.: A new version of Strang-Fix conditions. J. Approx. Theory 74, 221–225 (1993)

    MathSciNet  MATH  Google Scholar 

  21. Kazemi, M.M., Dehghan, M., Bastani, A.F.: On a new family of radial basis functions. J. Comput. A.pl. Math. 328, 75–100 (2018)

    MATH  Google Scholar 

  22. Maz’ya, V., Schmidt, G.: Construction of basis functions for high order approximate approximations, Mathematical Aspects of boundary elements methods (Palaiseau, 1998), Chapman and hall/CRC Res. Notes Math. 414, 191–202 (2000)

    Google Scholar 

  23. Maz’ya, V., Schmidt, G.: On quasi-interpolation with non-uniformly distributed centers on domains and manifolds. J. Approx. Theory 110, 125–145 (2001)

    MathSciNet  MATH  Google Scholar 

  24. Rabut, C.: An introduction to Schoenberg’s approximation. Comput. Math. Appl. 24, 139–175 (1991)

    MathSciNet  Google Scholar 

  25. Stein, E., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Univ. Press, Princeton (1971)

    MATH  Google Scholar 

  26. Schaback, R.: The missing Wendland functions. Adv. Comp. Math. 34, 67–81 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Strang, G., Fix, G.: A fourier analysis of the finite-element method. In: Geymonat, G. (ed.) Constructive Aspects of Functional Analysis, pp 793–840. C.I.M.E., Rome (1973)

  28. Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4, 389–396 (1995)

    MathSciNet  MATH  Google Scholar 

  29. Wu, Z.M.: Hermite-birkhoff interpolation of scattered data by radial basis functions. Approx. Theory. Appl. 8, 1–10 (1992)

    MathSciNet  MATH  Google Scholar 

  30. Wu, Z.M.: Multivariate compactly supported positive definite radial basis functions. Adv. Comput. Math. 4, 283–292 (1995)

    MathSciNet  MATH  Google Scholar 

  31. Wu, Z.M., Schaback, R.: Local error estimates for radial basis function interpolation of scattered data. IMA J. Numer. Anal. 13, 13–27 (1993)

    MathSciNet  MATH  Google Scholar 

  32. Wu, Z.M.: Operators on radial functions. J. Comput. Appl. Math. 73, 257–270 (1996)

    MathSciNet  MATH  Google Scholar 

  33. Wu, Z.M.: Compactly supported positive definite radial basis functions and the Strang-Fix condition. Appl. Math Comput. 84, 115–124 (1997)

    MathSciNet  MATH  Google Scholar 

  34. Wu, Z.M., Liu, J.P.: Generalized Strang-Fix condition for scattered data quasi-interpolation. Adv. Comput. Math. 23, 201–214 (2005)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors acknowledge the associate editor and the anonymous referee for insightful comments and valuable suggestions.

Funding

This work is financially supported by NSFC (11871074, 11501006, 61672032), NSFC Key Project (91330201,11631015), SGST (12DZ 2272800), Joint Research Fund by the National Natural Science Foundation of China and Research Grants Council of Hong Kong (11461161006), Fund of Introducing Leaders of Science and Technology of Anhui University (J10117700057) and the 4th Project of Cultivating Backbone of Young Teachers of Anhui University (J01005138), and Anhui Provincial Science and Technology Major Project (16030701091).

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Authors and Affiliations

  1. Department of Statistics, School of Economics, Anhui University, Hefei, People’s Republic of China

    Wenwu Gao

  2. Anhui Engineering Laboratory of Agro-Ecological Big Data, Anhui University, Hefei, People’s Republic of China

    Wenwu Gao

  3. Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, People’s Republic of China

    Wenwu Gao

  4. School of Mathematical Sciences, Fudan University, Shanghai, People’s Republic of China

    Xuan Zhou

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  1. Wenwu Gao
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Correspondence to Xuan Zhou.

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Gao, W., Zhou, X. Multiscale radial kernels with high-order generalized Strang-Fix conditions. Numer Algor 85, 427–448 (2020). https://doi.org/10.1007/s11075-019-00820-9

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